A generalization of CHSH and the algebraic structure of optimal strategies

David Cui1, Arthur Mehta1, Hamoon Mousavi2, and Seyed Sajjad Nezhadi2

1Department of Mathematics, University of Toronto, Toronto, Canada.
2Department of Computer Science, University of Toronto, Toronto, Canada.

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$\textit{Self-testing}$ has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick [26], iterated compression by Fitzsimons et al. [16], and NEEXP in MIP* due to Natarajan and Wright [27]. The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.

In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of LCS games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark [15]. Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal [11]. Additionally, our games have $1$ bit question and $\log n$ bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the $\textit{solution group}$ of Cleve, Liu, and Slofstra [10] to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered.

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